Bridge World Mystery # 1 Extended
The Bridge World's Mystery #1 was to determine the weakest possible trump holdings capable of making a slam or a grand slam against best defense, both when the opening lead is unrestricted and when an opening trump lead is barred. I will investigate the more general question of determining, for each bidding level from 1 to 7, exactly which trump holdings are strong enough to make a contract at that level, under each of the stated conditions on the opening lead.
For this purpose, we start by imposing a partial ordering on possible trump holdings. We will say that holding A is strictly weaker than holding B if B has at least as many cards as A, and the highest card in A is no higher than the highest card in B, the second highest card in A is no higher than the second highest card in B, and so on, the lowest card in A being likewise no higher than the corresponding card in B. For instance A Q 9 is strictly weaker than either A Q 9 2 or A Q 10, but we don't attempt to say whether A Q 9 2 is weaker or stronger than A Q 10.
The following table gives the final results of the investigation. The meaning of the table is that a trump holding is capable of making a contract at the given level under the given condition on opening lead, if and only if the holding either appears in the table or is strictly stronger than a holding in the table.
Level 
Opening lead unrestricted. 
Trump lead barred. 
1 
2 
2 
2 
4 3 2 
4 3 2 
3 
6 5 4 3 2 
6 5 4 3 2 
4 
8 7 6 5 4 3 2 
8 7 6 5 4 3 2 
5 
10 9 8 7 6 5 4 3 2 
10 9 8 7 6 5 4 3 2 
6 
Q J 10 9 8 7 6 5 4 3 2 
Q J 10 9 8 7 6 5 4 3 2 
7 
A J 10 9 8 7 6 5 4 3 2 
A J 10 9 8 7 6 5 4 3 2 
There are some general principles which will make the analysis easier. Speaking roughly, if one of the defenders holds more trumps than the number of tricks declarer can afford to lose, then declarer will need to have some highranking trumps in order to succeed. The simplest example is when declarer is in a grand slam; then if either defender holds a trump, declarer will surely need the A of trumps. Less trivially, if one of the defenders holds 2 or more trumps, and declarer must make a small slam, then the defenders cannot have both the A and K of trumps, since at double dummy, they will always be able to avoid crashing their honors.
Explicitly, the general principle involved is:
A) If declarer can only afford to lose k tricks, and one of the defenders holds n or more trumps, where n is greater than k, then
(i) declarer's trump holding must include at least nk of the top 2nk1 trumps; and
(ii) declarer must win at least nk tricks with trumps.
To see that (i) is true, suppose that declarer holds only nk1 of the top 2nk1 trumps; then the defenders hold n of the top 2nk1 trumps between them. Since at least one of the defenders holds n or more cards in the trump suit, the defenders can always arrange to play their n highest trumps on n different tricks; declarer can contribute high trumps to at most nk1 of those tricks, and so must lose at least k+1 of them, thus going set.
As for (ii), declarer must win at least 13k tricks; since one of the defenders will be playing trumps on at least n different tricks, declarer can win at most 13n tricks in the side suits, so declarer must win at least nk tricks with trumps.
It turns out that for trump holdings of 6 or more cards, principle A(i) alone already settles the question of which holdings can make which contracts; for shorter trump holdings, however, more detailed analysis will be required. Here is a template to show how to make the appropriate contract with any of the 6card holdings listed in the table (in this and all subsequent tables, spades will be trumps):

This template covers all cases of 6card trump holdings. 7level, South holds A Q 10 8, East holds K J 9. 6level, South holds K J 9 4, East holds A Q 10. 5level, South holds Q 10 5 4, East holds A K J. 4level, South holds J 6 5 4, East holds A K Q. In each case, declarer leads trumps twice from the North hand to neutralize East's trump holding, then finishes drawing trumps and cashes 9 side suit winners. 
It is straightforward to construct a similar template for 7card and longer holdings from the table. For this reason, we will ignore 6card and longer holdings in the following analysis.
As a sort of converse to A, we also have the following general principle:
B) If neither defender holds more trumps than the number of tricks declarer can afford to lose, then declarer can succeed even with all small trumps.
To see this, simply set the defensive hands so that every suit splits as evenly as possible, and give declarer lots of winners in the side suits. Then the defenders will make only trump tricks, and since they will be unable to make their trumps separately, and neither defender has enough trumps to defeat declarer, declarer must be able to succeed.
We now proceed level by level. In each case (except the 1level, which is very simple) we begin with a preliminary analysis based on the general principles, and then proceed to specific considerations in the two cases (a) Opening lead unrestricted and (b) Opening trump lead barred.
This is easy; any trump holding except a double void can succeed at a llevel contract. If declarer has no trumps in either hand, then one of the defenders will have at least 7 trumps, and the defenders will take at least 7 trump tricks; but if declarer has as much as the singleton 2 of trumps, then we can arrange the defensive hands with a 66 trump split and give declarer 7 cashing tricks in the side suits; and this works whether or not the opponents are allowed to lead trumps (this is an example of principle B).
Preliminary analysis: Principle B says that any three trumps suffice to make a 2level contract, so suppose declarer has only 1 or 2 trumps. Then one of the defenders will have at least 6 trumps, and declarer can only afford to lose 5 tricks, so we can apply principle A(i) to see that declarer needs at least 65 of the top 2x651 trumps, i.e., 1 of the top 6; in other words, declarer must have at least the 9 of trumps. Furthermore, principle A(ii) says that declarer needs to win at least 1 trick with a trump.
(a) Opening lead unrestricted.
If the defenders are allowed to lead trumps, then declarer needs at least A or K2 of trumps to prevent the defenders drawing trumps; together with 4 3 2, these holdings are good enough.
(b) Opening trump lead barred.
When the defenders are prevented from drawing declarer's trumps at the outset, declarer can succeed with K or with 9 2; but Q alone is not good enough. We give the demonstration of this last statement in considerable detail, as a model for later demonstrations.
Suppose declarer's trump holding consists of Q alone. We know that declarer needs to win a trick with the Q of trumps, and will also need to win 7 tricks in the side suits. This means the defenders must both follow suit while declarer cashes 7 side winners, so the opposing trumps must split 66. Clearly declarer can't score a trump trick by leading the Q, so the Q must score as a ruff. Furthermore, this ruff must occur after declarer has already secured the 7 tricks in the side suits (if declarer successfully ruffs with the Q while the defenders still have sidesuit cards, then the defenders will no longer have to follow suit while declarer cashes 7 side winners). But if declarer tries ruffing with the Q after cashing 7 side winners, then both defenders will be reduced to their 6 trumps, and they will be able to ruff high and play the other high trump, preventing the Q from scoring a trick.
Preliminary analysis: Principle B says that any 5 trumps suffice to make a 3level contract. If declarer has 3 or 4 trumps, then the defenders have 9 or 10 trumps, so at least one of the defenders has at least 5 trumps, so Principle A says that declarer needs at least 1 of the top 5 trumps, and must win at least 1 trick with a trump. If declarer has 1 or 2 trumps, then the defenders have 11 or 12 trumps, so at least one of the defenders has at least 6 trumps, so we can apply A(i) twice to see that declarer needs at least 1 of the top 5 trumps and at least 2 of the top 7 trumps; in particular, 1 trump is never good enough, and if declarer has 2 trumps, then A(ii) says that declarer must win tricks with both trumps.
(a) Opening lead unrestricted.
As noted, 6 5 4 3 2 suffices.
With 4 trumps, declarer needs the J to prevent the defenders drawing trumps, and J 4 3 2 is good enough. (Note: of course, if the defensive trumps were, for instance, A K Q opposite J 9 8 7 6 5, while declarer held 10 4 3 2 all in one hand, then the defenders would be unable to draw all of declarer's trumps, but the 63 split would still be fatal to a 3level contract; defenders would draw three rounds of trumps, and declarer would be able to score at most 7 side winners and 1 trump trick. Similarly, declarer would be helpless against blocked defensive trump holdings split 72, 81, 90. Such observations will be taken for granted in the remaining analyses.)
With 3 trumps, declarer needs the Q to prevent the defenders drawing trump, and Q 3 2 is good enough.
With 2 trumps, declarer needs the A (otherwise the defenders can draw at least one round of trumps, preventing declarer from scoring both trumps); the second trump must be at least the 8, since declarer needs 2 of the top 7 trumps, and A 8 is indeed good enough.
(b) Opening trump lead barred.
With 4 trumps, declarer needs at least the 10 (1 of the top 5) and 10 4 3 2 is good enough.
With 3 trumps, J 3 2 is good enough, and so is 10 9 2; but 10 8 7 is not good enough. Again, we demonstrate this last claim in some detail to serve as a model for further discussions.
Suppose declarer has 10 8 7 and the opening trump lead is barred. Then declarer cannot succeed by cashing 8 side winners and scoring a subsequent trump (defenders could ruff high and still draw declarer's trumps  compare with the argument above showing that Q alone cannot make a 2level contract); suppose instead that declarer succeeds with only 7 side winners and 2 ruffs. Then at the trick on which declarer attempts to score a second ruff, one of the defenders must still be following suit, and that defender must hold the A K Q J of trumps (otherwise, the defenders could certainly ruff high and draw declarer's remaining trumps). Then the other defender's initial trump holding must have been 9 6 5 4 3 2 (the defender with A K Q J has followed to 9 side tricks, and so cannot have any other trumps). The full play is then to cash 7 side tricks, then score two ruffs while the A K Q J still follows suit and the 9 6 5 4 3 2 underruffs or is overruffed. The first ruff must be taken in the hand lying over the 9 6 5 4 3 2 hand (otherwise, declarer would have to ruff with the 10 to shut out the 9, and then the long trump hand could win the next trick with the 9 of trumps and lead another trump to draw declarer's remaining trumps); and that hand must contain the 10 (otherwise the 9 will win and draw trumps) and at least one other trump (otherwise declarer must expend the 10 on the 8th trick, allowing the 9 to win the 9th trick and draw trump); but now after overruffing at trick 8, declarer is in the wrong hand and cannot score the 10. A similar but simpler argument shows that declarer cannot succeed with 6 side winners and 3 ruffs.
With 2 trumps, the K 8 is good enough, but Q J is not (in the endgame, the defenders ruff high and cash the other high trump to hold declarer to 1 trump trick).
Preliminary analysis: Principle A says: with 5 trumps, declarer needs 1 of the top 4 trumps, and must take at least 1 trump trick; with 3 or 4 trumps, declarer needs 1 of the top 4 and 2 of the top 6, and must take at least 2 trump tricks; with 1 or 2 trumps, declarer never succeeds.
(a) Opening lead unrestricted.
With 5 trumps, J 5 4 3 2 is the theoretical minimum, and is in fact good enough.
With 4 trumps, declarer needs at least the Q to prevent the defenders drawing trump, and the second card must be at least the 9, and Q 9 3 2 is good enough.
With 3 trumps, declarer needs at least the K to prevent the defenders drawing too many trumps; K 10 2 is good enough, but A 9 8 is not (the defenders lead trumps, and continue if declarer ducks. Declarer must win the first or second trump, and then declarer must either (i) win 8 side tricks and then score a second trump trick, which will be impossible, since both defenders will be down to trumps only, and the defenders will have enough high trumps left to ruff high and pull declarer's last trump(s); or (ii) win 7 side tricks and then score 2 further trump tricks, which is also impossible, since after 1 ruff, both defenders will again be down to trumps only, and they can afford to ruff the next trick high and draw declarer's last trump).
(b) Opening trump lead barred.
Now with 4 trumps, J 9 3 2 (the theoretical minimum according to A(i)) is good enough.
With 3 trumps, K 9 8 is now good enough, but Q J 10 and A 9 7 are not (for Q J 10, whether declarer tries to score 8 side tricks and 2 ruffs, or 7 side tricks and 3 ruffs, the defenders will wait until they are in a position to overruff one of declarer's honors, and will then be able to draw the rest of declarer's trumps; for A 9 7, if declarer tries to (i) score 8 side tricks and 2 trump tricks, then the defenders can ruff high at trick 9, and if declarer discards or underruffs, the defenders play another high trump, and continue with a third high trump if declarer ducks again; in all cases, after declarer wins the trump A, the defenders are in a position to win the next trick as cheaply as possible and draw the rest of declarer's trumps; or (ii) score 7 side tricks and 3 trump tricks, then the defenders' trumps must split 64, with the K Q J 10 in the short hand; and an argument similar to the 10 8 7 argument above shows that declarer can't score all three trumps).
Contracts at the 5level and higher are considerably more complicated to analyse, particularly for holdings of 5 and 4 trumps, because of the proliferation of cases to be considered. We begin with two studies of features common to 5, 6 and 7level contracts, which will settle most of the critical cases. We will refer to these later as the bulk analysis. With 5 trumps, most of the critical cases arise when the defenders have enough top trumps to hold declarer to the contract, and declarer must neutralize the defenders' intermediate trumps.
5card trump holdings in highlevel contracts
Suppose declarer can afford to lose k tricks, where k is 0, 1, or 2, and the defenders hold 8 trumps including the k highest trumps, while declarer holds the next highest trump. We consider three cases:
(i) Defenders hold the highest of the other trumps; that is, at the 5level the defenders hold A K J, at the 6level they hold A Q, or at the 7level they hold K.
(ii) Defenders hold the second and third highest of the other trumps; that is, at the 5level they hold A K 10 9, at the 6level they hold A J 10, or at the 7level they hold Q J.
(iii) k=1 or 0, and defenders hold either the second or the third highest, together with both the fourth and fifth highest of the other trumps; that is, at the 6level they hold A J 9 8 or A 10 9 8, or at the 7level they hold Q 10 9 or J 10 9. (Note: the pattern is disrupted for 5level holdings in case (iii) because when the defensive holding is 8 trumps to the A K 9 8 7 and the defensive trumps split 44, declarer can force the defenders to expend two of their significant trumps on the same trick.)
Our conclusion will be that in most of these cases, declarer cannot succeed. The only exceptions are:
(I) At the 7level, declarer with A K Q 8 6 (or better) can succeed if one defender has J 10 9 alone.
(II) At the 6level, declarer with K Q J 7 5 (or better) can succeed if one defender has 10 9 8 alone.
(III) At the 5level, if an opening trump lead is barred, declarer with Q 10 9 3 2 (or better) can succeed if one defender has A K J and the other defender has the 5 remaining trumps.
(IV) At the 7level, if an opening trump lead is barred, declarer with A K J 8 7 (or better) can succeed if one defender has Q 10 9 and the other defender has the 5 remaining trumps.
(V) At the 6level, if an opening trump lead is barred, declarer with K J 10 9 7 (or better) can succeed if one defender has A Q and the other defender has the 6 remaining trumps.
To see that all of this is true, we consider in turn each possible splitting of the defensive trumps. First, suppose the defensive trumps split 44. Then it is not hard to see that the defenders can always make enough trump tricks by judicious timing; in effect, declarer needs one trump lead and two trump coups, and is an entry short. We work out one example in detail; all the other cases are similar.
Suppose the defenders hold A J 10 against a 6level contract. If declarer leads an early round of trumps ("early" meaning before all the side winners are cashed), then declarer's trumps must be split 41 (if they were 32, the defenders would win the ace and play another trump; if they were 50, a trump lead would set up a second trump trick for the defense). The defenders will win the A and exit safely in a side suit. Declarer can't lead a second round of trumps, so must cash all remaining side winners, finishing in the short trump hand after ten tricks (one round of trumps and nine side winners) and play a plainsuit card. The defenders' J 10 combination ensures that declarer will have to expend the K or Q to win trick eleven, and now declarer will be stuck in hand, leading away from the remaining honor while the defenders still have at least one of their honors.
Suppose then that there is no early trump lead. After nine tricks both defenders will be reduced to 4 trumps. To make player references simpler, suppose North leads to trick 10. North can't lead the K or Q (the A would win, and the J 10 combination would take another trick), so North must lead a small trump or a side card. If West has either the J or 10, then South will have to play the K or Q, leaving the defenders with A J 10 and declarer with only one honor, so East must have both the J and 10. In that case, East plays the 10, forcing South to play the K or Q, which is allowed to hold. Now South leads to trick 11, and declarer must play the other honor to avoid losing to East's J, but this leaves the defenders with the A and J for the last two tricks.
Next, suppose the defensive trumps split 53.
(a) Suppose declarer succeeds by leading an early round of trumps, and consider first the case of a 7level contract. Then all declarer's trumps must be in the same hand (otherwise the long defensive trump holding would now have more trumps than declarer's combined holding). Declarer clearly can't lead away from a holding headed by the A Q or A K 10 without setting up a quick trick for the defenders, and a lead from A K J is also futile, so declarer must have A K Q, defenders J 10 9. Now after declarer cashes a high trump, we can apply principle A(i) to the remaining 10 trumps (remembering that one of the defenders will still have 4 trumps) to see that declarer must hold the highest, two of the 3 highest, 3 of the 5 highest, and 4 of the 7 highest remaining trumps. In particular, the defenders can't still have the J, 10, and 9 of trumps (else declarer would have only 2 of the 5 highest remaining trumps), so the defenders must be forced to play one of their high trumps on the first round, which can only happen if the 3card holding is precisely J 10 9. Then in order to have 3 of the 5 highest and 5 of the 7 highest remaining trumps, declarer needs to have at least 8 6 in addition to the A K Q. This is Exception (I); the play is for declarer to cash the three top trumps, then cash 8 side winners finishing in the hand opposite the remaining trumps, and use the 8 6 to coup the 7 5.
A precisely similar argument at the 6level gives Exception (II). (Note: at the 5level, the corresponding exception would involve a defensive holding of 9 8 7 opposite five trumps to the A K, but as noted in the beginning, we are not now considering this particular defensive holding).
(b) When the defensive trumps split 53 and declarer does not lead an early round of trumps, declarer plans to win 8 side tricks and then score enough trump tricks for the contract on some kind of crossruff. If the 5card defensive holding includes a trump winner, it can interrupt the crossruff by overruffing as soon as possible and leading another trump (if both declarer and dummy still have trumps at this point, this will take out two of declarer's trumps on one trick, while if all declarer's remaining trumps are in one hand, that hand will now be endplayed  the only exception is at the 6level when the 3card defensive holding is J 9 8 or 10 9 8 alone; with J 9 8, the defence succeeds by holding off the A as long as possible, while 10 9 8 just leads to a weaker form of Exception (II)). Also, if the 5card holding includes one of the intermediate trumps, then declarer, while crossruffing, will be forced to play a high trump on air, promoting the defensive holding. So all the significant defensive trumps must lie with the 3card holding, and declarer takes 8 side tricks and crossruffs 2 more tricks to lead to a 3card ending in which both defensive hands are down to all trumps. The analysis then proceeds as in the case of a 44 split, and leads to only two new cases in which the declarer can succeed. At the 5level, when the 3card holding is A K J, declarer can cash 8 tricks, crossruff for two more while the A K J follows suit, and finally score the Q en passant. For this to work, declarer needs to be able to outruff the long trump hand twice without using the Q, so declarer needs at least the 10 and 9 of trumps, and Q 10 9 3 2 is good enough, providing that the opening trump lead is barred. This is Exception (III). The other case is at the 7level, when the 3card holding is Q 10 9; again, declarer cashes 8 tricks, crossruffs for two more while the Q 10 9 follows suit (declarer needs the 8 7 in order to outruff the 5card holding), and crossruffs for 3 more tricks with the A K J, scoring the J behind the Q. This is Exception (IV), and finishes our consideration of the case of a 53 split.
It is easy to see in each of our cases that declarer can never prevail against 80 or 71 trump splits, so we are left with the 62 split to consider.
Declarer obviously can't succeed at a 7level contract with only 5 trumps when one defender has 6 trumps, so we need only consider 5 and 6level contracts. Again, it is possible to show that if the long defensive trump holding includes any of the significant trumps, that hand will be able to disrupt the necessary crossruff, so all the significant defensive trumps must lie in the 2card holding. The only possibility is then for the short holding to be A Q alone at a 6level contract. With only 7 side winners, declarer must score all 5 trumps separately (in particular, the trump lead must be barred). The play must then be to cash 7 side winners, crossruff 4 tricks while the A Q holding still follows suit, and finish by scoring the trump K en passant. Declarer must be able to outruff the 6card holding 4 times, so the K must be accompanied by two trumps both higher than any of the trumps in the 6card holding, and the opposite hand must be able to overruff the two highest trumps in the 6card holding. This means declarer's trumps must be at least K J 10 9 7, which leads to Exception (V).
This completes our (lengthy) preliminary examination of 5card trump holdings at high level contracts. We deal more briefly with the case of 4card trump holdings. Here the defenders will always score at least one long trump trick, and most of the critical cases occur when the defenders can hold declarer to the contract in top trumps and long trumps alone, and declarer must neutralize the defenders' intermediate trumps.
4card trump holdings in highlevel contracts
Suppose declarer can afford to lose k tricks, where k is 1 or 2 (4 trumps cannot make a grand slam), and the defenders hold 9 trumps including the k1 highest trumps, while declarer holds the next highest trump. This time we consider only two cases:
(i) Defenders hold the highest of the other trumps; that is, at the 5level the defenders hold A Q, or at the 6level they hold K.
(ii) Defenders hold the second and third highest of the other trumps; that is, at the 5level they hold A J 10, or at the 6level they hold Q J.
Our conclusion will again be that in most of these cases, declarer cannot succeed. The only exceptions are:
(I) At the 6level, if an opening trump lead is barred, declarer with A Q 10 7 (or better) can succeed if one defender holds K J 9 8 and the other defender has the 5 remaining trumps.
(II) At the 5level, if an opening trump lead is barred, declarer with K J 8 2 (or better) can succeed if one defender holds A Q 10 9 and the other defender has the 5 remaining trumps.
(III) At the 5level, if an opening trump lead is barred, declarer with K 10 9 7 (or better) can succeed if one defender holds A Q J and the other defender has the 6 remaining trumps.
To see that all of this is true, we consider in turn each possible splitting of the defensive trumps. This time we only need to consider the 54 split, and in the case of the 5level, the 63 split.
Suppose the defenders trumps split 54, and declarer is in a small slam. Then declarer, with only 8 side tricks, must score tricks with all 4 trumps. In particular, if the defenders are allowed to lead a trump, then declarer's trumps must all lie in one hand. Declarer, who must lose a trick to the long trump in any case, can't afford to also lose a trick to the defender's high trumps. If declarer's trumps are all in one hand, this will be impossible to avoid  the best declarer can do is to achieve the equivalent of one trump coup and one endplay, but that will still leave a high trump loser. Consider in detail the case of a defensive holding including the K. Declarer can't lead trumps, so cashes 8 side tricks, and then leads a plain suit from the hand with no trumps. The defenders allow declarer to win this trick, and declarer still can't afford to lead a trump, so declarer exits with the last side suit card. The defenders ruff this low and are endplayed, forced to allow declarer to score another trump trick, but now at trick 12 declarer is finally forced to lead away from the trump A.
Suppose then that declarer's trumps are split between the two hands. As we have seen, this means that the trump lead must be barred. Declarer will cash 8 side tricks, and then try to score all four trumps by ruffing. The bynow familiar style of argument shows that this cannot succeed if the defenders hold the A or the K Q or the Q J 10 of trumps, so declarer's holding must be headed by the A Q 10 at least, and principle A(i) indicates that declarer's fourth trump must be at least the 7. This gives Exception (I), where declarer holds A Q 10 over K J 9 8 and opposite the 7, and cashes 8 side tricks, ruffs with the 7 while the K J 9 8 hand still follows suit, then overruffs cheaply and exits with a side card to endplay the defender with the high trumps.
Next, consider a 5level contract with a 54 split. The analysis is similar; declarer must score 8 side tricks and 3 trump tricks, so if the defenders are allowed to lead trumps, all of declarer's trumps must lie in the same hand, and in this case, declarer will of necessity lose the A of trumps, another high trump, and a long trump. Suppose instead that declarer's trumps are split between the two hands and the trump lead is barred. Declarer cashes 8 side tricks before embarking on a crossruff, which is doomed to fail if the defenders hold the A K or the A Q J, so declarer's trumps must be at least K J, and principle A(i) says declarer's third trump must be at least the 8. This leads to Exception (II), where declarer holds K J 2 over A Q 10 9 and opposite the 8, and cashes 8 side tricks, ruffs with the 8 while the A Q 10 9 follows suit, then leads a plain suit through the A Q 10 9 and has an answer to whatever the defender tries.
Finally, consider a 5level contract with a 63 split. The trump lead must be barred (otherwise the defenders play two rounds of trumps, leaving declarer with at least 3 trump losers), and declarer can't play trumps for the same reason, so declarer must cash 7 side winners and then score all four trumps separately. As usual, this crossruff must fail if the long trump hand has any significant trumps, but since declarer can ruff three times while the 3card holding is still following suit, declarer can succeed with a trump holding as weak as K 10 over A Q J, with 9 7 opposite; declarer cashes 7 side winners finishing in the hand with K 10, then crossruffs for three tricks while the 6card holding alternately is overruffed and underruffs, then scores the K en passant. This is Exception (III).
We now return to the levelbylevel analysis, using the results just obtained.
Preliminary analysis: Principle A says: with 5 trumps, declarer needs 1 of the top 3 and 2 of the top 5, and must take at least two trump tricks; with 3 or 4 trumps, declarer needs 1 of the top 3, 2 of the top 5, and 3 of the top 7, and must take at least three trump tricks.
(a) Opening lead unrestricted.
With 5 trumps, K 10 4 3 2 and Q J 9 3 2 are both good enough, but the bulk analysis shows that Q 10 9 8 7 and Q J 8 7 6 both fail.
With 4 trumps, K Q 10 7 and A J 8 2 suffice, but K J 10 9, K Q 9 8, K Q J 6, and A 10 9 8 all fail. (The bulk analysis shows that K J 10 9 and K Q 9 8 fail. For K Q J 6, a trump lead will beat this unless all four offensive trumps are in the same hand; then declarer must find the defensive trumps 54 and must score 3 trump tricks to go with 8 side tricks; but the 5card holding must include at least one card higher than the 6, and the usual sort of analysis shows that the defense can prevent the 6 scoring a trick while ensuring that the A captures one of the top cards. For A 10 9 8, the defenders lead trumps to knock out the ace, and then wait for a chance to either win a small trump or overruff one of declarer's intermediate trumps, after which they can cash their remaining high trump(s), leaving declarer with only 2 trump tricks.)
With 3 trumps, A Q 9 suffices, but A J 10 and A K 8 fail (declarer must find the defensive trumps 55, and must score tricks with all 3 trumps. For A J 10, the defenders can always overruff one of declarer's intermediates, unless the A J 10 are all behind the K Q; but then, after declarer cashes 8 side tricks, the defender with the high trumps ruffs the next trick low, allowing the 10 to score, but forcing declarer to lead away from the hand with the A J remaining; declarer must lead a side card, but then the defenders ruff low and play a high trump to knock out the A, and declarer has no way of scoring the J. For A K 8, declarer's best try in the endgame is to discard when the defenders ruff up with an intermediate trump; the defenders then play another intermediate, which declarer must win; declarer again exits with a side loser, which is ruffed low, and another intermediate trump leaves declarer leading away from the 8 at trick 12).
(b) Opening trump lead barred.
With 5 trumps, Q 10 9 3 2 suffices, but Q 10 8 7 6 still fails as shown by the bulk analysis.
With 4 trumps, K 10 9 7, A 10 8 2, and K J 8 2 all suffice, while Q J 10 9, K 10 8 7, and K 10 9 6 all fail (The last two facts follow from the bulk analysis. For Q J 10 9, declarer must score 3 trump tricks, so the defenders wait until they can overruff one of declarer's cards, and then play their other high trump to hold declarer to 2 trump tricks.)
With 3 trumps, the analysis in (a) did not depend on an opening trump lead, so A Q 9 is still the minimum.
Preliminary analysis: Principle A says: with 5 trumps, declarer needs 1 of the top 2, 2 of the top 4, and 3 of the top 6, and must take at least three trump tricks; with 3 or 4, declarer needs 1 of the top 2, 2 of the top 4, 3 of the top 6, and 4 of the top 8 and must take at least four trump tricks (in particular, 3 trumps never succeed).
(a) Opening lead unrestricted.
With 5 trumps, K Q J 7 5, K Q 10 8 2, A Q 9 3 2, and A J 9 8 2 are good enough, but K J 10 9 8, K Q 9 8 7, K Q 10 7 6, K Q J 6 5, K Q J 7 4, A J 10 7 6 all fail (This follows from the bulk analysis for the first five holdings. For A J 10 7 6, a separate analysis is required; declarer's best try is to have A J 10 7 behind K Q 9 8; then if declarer leads an early trump, the defender plays the Q; the defender later ruffs with the 8, forcing declarer to overruff and lead from the J 7 into the K 9 or from the A J into the K 9, depending on whether declarer ducked or won the earlier trump lead).
With 4 trumps, A K J 7 suffices, but A Q J 10 and A K 10 9 fail, as shown in the bulk analysis.
(b) Opening trump lead barred.
With 5 trumps, K J 10 9 7 suffices, as shown by the bulk analysis.
With 4 trumps, A Q 10 7 suffices, as shown by the bulk analysis.
Preliminary analysis: Principle A says: with 5 trumps, declarer needs the ace, 2 of the top 3, 3 of the top 5, and 4 of the top 7; and of course, 4 or fewer trumps never succeed.
(a) Opening lead unrestricted.
With 5 trumps, A K Q 8 6 and A K J 9 2 suffice, while A Q J 10 9, A K 10 9 8, A K Q 8 5, A K J 8 7 all fail, as shown by the bulk analysis.
(b) Opening trump lead barred.
With 5 trumps, A K J 8 7 suffices, while A K J 8 6 fails, as shown by the bulk analysis.
The following tables give templates showing how to make contracts with the minimal trump holdings determined in the preceding analysis. In every case, spades are trumps.

This template covers 2 cases when the trump lead is permitted: 2level, South holds A, East x x x x x x. 1level, South holds 2, East x x x x x x. Declarer cashes 7 side winners; at the 2level, the A of trumps remains as the 8th trick. This template also covers 1 case when the trump lead is forbidden: 2level, South holds K, East A x x x x x. Declarer cashes 7 side winners, finishing in the North hand, and leads a heart to score the trump K en passant. 

This template covers 2 cases when the trump lead is permitted: 3level, South holds A 8, East K Q J 10 9. 2level, South holds K 2, East A x x x x. Declarer cashes 7 side winners, finishing in North, then ruffs a heart while East follows suit and West underruffs; this suffices at the 2level, while at the 3level the A of trumps remains as the 9th trick. 

This template covers 2 cases when the trump lead is forbidden: 3level, South holds K, North 8, East A Q J 10 9. 2level, South holds 2, North 9, East A K Q J 10. Declarer cashes 7 side winners, finishing in the South hand, then ruffs a heart in the North hand while West is overruffed and East follows suit. In the case of the 3level contract, declarer finishes by leading a diamond from the North hand to score the trump K en passant. 

This template covers 4 cases when the trump lead is permitted: 5level, South holds A Q 9, East K J 10 x x. 4level, South holds K 10 2, East A Q J x x. 3level, South holds Q 3 2, East A K x x x 2level, South holds 4 3 2, East x x x x x. The trickiest case is when South needs 2 tricks from K 10 2. If the defenders play on trumps, South wins as quickly and cheaply as possible, cashes 8 side winners, finishing in North, then leads a heart to score a second trump trick en passant. If defenders don't lead trumps, declarer cashes 8 side winners, finishing in North. Now a heart lead from North; if East ruffs with the A, declarer discards, and then ducks if East continues with the Q; if East ruffs with the 8 or 9, declarer overruffs with the 10 and exits with a club, scoring a later trick with the trump K; if East ruffs with the J or Q, declarer discards a club, and if East now leads the A, declarer is left with a trump tenace, while if East leads any other trump, declarer wins cheaply and exits with a club, eventually scoring a second trump trick. This template also covers 1 case when the trump lead is forbidden: 3level, South holds J 3 2, East holds A K Q x x. Here South cashes 8 tricks finishing in North, then leads a heart to score the trump J en passant. 

This template covers 2 cases when the trump lead is forbidden: 4level, South holds K 9, North 8, East A Q J 10. 3level, South holds 10 2, North 9, East A K Q J. Declarer cashes 7 side winners, finishing in the North hand, then crossruffs a heart and a diamond while East still follows suit and West underruffs and is overruffed. At the 4level, another heart allows South to score the K en passant. 

This template covers 5 cases when the trump lead is permitted: 6level, South has A K J 7, East Q 10 9 8. 5level, South has K Q 10 7, East A J 9 8. 5level, South has A J 8 2, East K Q 10 9. 4level, South has Q 9 3 2, East A K J 10. 3level, South has J 4 3 2, East A K Q x. In the last case, declarer easily scores a trick with the trump J, so suppose we are in one of the other 4 cases. If the defenders lead trumps, South wins as quickly and cheaply as possible. Declarer then cashes 8 side winners, finishing in North, and ruffs a heart with the 7, 8, or 9 while West underruffs and East follows suit. Declarer then exits with a club and waits for the remaining trump trick(s). 

This template covers 5 cases when the trump lead is forbidden: 6level, South has A Q 10, North 7, East K J 9 8. 5level, South has K J 2, North 8, East A Q 10 9. 5level, South has A 10 2, North 8, East K Q J 9. 4level, South has J 3 2, North 9, East A K Q 10. 3level, South has 4 3 2, North 10, East A K Q J. Declarer cashes 8 side winners, finishing in the South hand, then ruffs a heart with North's trump while West is overruffed and East follows suit. If declarer needs further trump tricks, a diamond lead from North, possibly followed by a club exit later, will produce the needed tricks. The trickiest case is A 10 2; here if East ruffs high at trick 10, South must discard a club. East returns a second high trump, and South ducks to retain a trump tenace and endplay East. 

This template covers 1 case when the trump lead is forbidden: 5level, South holds K 10, North 9 7, East A Q J. Declarer cashes 7 side winners, throwing clubs from South on North's heart winners and finishing in South, then crossruffs two diamonds and a heart while East follows suit and West is overruffed twice and underruffs once. Now a further heart lead from the North hand scores the K en passant. 

This template covers 6 cases when the trump lead is permitted: 7level, South holds A K J 9, East Q 10 x x. 6level, South holds K Q 10 8, East A J 9 x. 5level, South holds Q J 9 3, East A K 10 x. 5level, South holds K 10 4 3, East A Q J x. 4level, South holds J 5 4 3, East A K Q x. 3level, South holds 6 5 4 3, East x x x x. In each case, North leads the trump 2 early on, and if East doesn't play a winner, South just covers East's card. Later, declarer cashes 9 side winners finishing in the North hand, then leads a heart to promote the necessary trump trick(s). The trickiest case is when South holds K 10 4 3; then if East wins the A on the first round and later plays the J on the lead of the fourth heart from North, declarer must underruff to promote the K 10 into a tenace over the Q 9 and leave East on lead. 

This template covers 2 cases when the trump lead is permitted: 7level, South holds A K Q 8 6, East 7 5 4 3 2. 6level, South holds K Q J 7 5, East A 6 4 3 2. In each case, South leads the three top trumps, swallowing West's holding. Then declarer cashes 8 side winners, finishing in North for a trump coup; at the 7level, South will hold 8 6 over East's 7 5, while at the 6level South will hold 7 5 either over East's 6 4 (if East has taken the A) or over East's A 6 (if East has ducked 3 times). 

This template covers 1 case when the trump lead is permitted: 6level, South holds A J 9, North 8 2, East K Q 10. North leads the trump 2 early, and South covers whichever card East plays. Declarer cashes 8 side winners finishing in the North hand, then crossruffs a heart and a diamond while West ruffs with low trumps and East follows suit. Now a heart lead from the North hand allows declarer's last trump to score. This template also covers 2 cases when the trump lead is forbidden: 7level, South holds A J 8, North K 7, East Q 10 9. 5level, South holds Q 10 3, North 9 2, East A K J. Declarer cashes 8 side tricks, finishing in North, then crossruffs a heart and a diamond (ruffing with the 8 and 7 at the 7level, the 10 and 9 at the 5level). At the 7level, declarer finishes with a high crossruff, while at the 5level, one further heart lead is enough to promote the trump Q en passant. 

This template covers 1 case when the trump lead is permitted: 6level, South holds A Q 9, East K J 10 8, North 3 2. North leads the trump 2 early, and South covers whichever card East plays. If East allows the 9 to win, declarer later scores the A and Q with a simple trump coup, so suppose East forces the A or Q by ruffing with the K or 10 respectively. Declarer then cashes 9 side winners, finishing in the North hand, and leads a heart. If East again ruffs high, South discards a club to promote a trump tenace and endplay East. 

This template covers 1 case when the trump lead is forbidden: 6level, South holds K J 10, North 9 7, East A Q. Declarer cashes 7 side winners (throwing clubs from the South hand on North's heart winners), finishing in North, then crossruffs heart, diamond, club, diamond, West alternately underruffing and being overruffed while East follows suit. Now a club lead from North allows the trump K to score en passant. 